A dishonorable discharge was a form of separating a member of the Federation Starfleet from the service. This form of separation was reserved for those members who had committed serious crimes or violations of regulations. Such discharges were considered shameful by those in Starfleet.
Doctor Julian Bashir was dishonorably discharged from Starfleet following a court martial for his actions in accessing the Shedai meta-genome. Andorian President Kellessar zh'Tarash quickly pardoned him and managed to have Bashir's discharge changed to honorable. (ST - The Fall novel: Peaceable Kingdoms; ST - Section 31 novel: Disavowed)
In an alternate timeline created by Dr. Georges Mordreaux's time travel device, Starfleet forced Captain Hunter of the USS Aerfen to accept a dishonorable discharge in order to keep Mordreaux's work secret. She joined the Free Commandoes and was killed in action a few months later. (TOS novel: The Entropy Effect)